The Optical Hall Effect (OHE) is a physical phenomenon which describes the occurrence of transverse and longitudinal magnetic field-induced birefringence, caused by the nonreciprocal magneto-optic response of electric charge carriers. The term (OHE) is used since the classic electrical Hall Effect (HE), and the (OHE) effect both find explanation within the Drude model. The term Optical Hall Effect (OHE) is used in analogy to the classic electrical Hall Effect as the electrical Hall effect and certain cases of (OHE) observation can be explained by extensions of the classic Drude model for the transport of electrons in matter, (eg. Metals). For the (OHE), Drude's classic model is extended by a magnetic field and frequency dependency, describing the electron's momentum under the influence of the Lorentz force. As a result an anti-symmetric contribution is added to the dielectric tensor, the sign of which depends on the type of the free charge carrier (electron or hole). The non-vanishing off-diagonal elements of the dielectric tensor reflect the magneto-optic birefringence, which lead to conversion of [p]-polarized into [s]-polarized electromagnetic waves, and vice versa. The (OHE) allows determination of concentration, mobility, and effective mass of the free electrons as the (OHE) can be quantified in terms of the Mueller matrix, which characterizes the transformation of an electromagnetic wave's polarization state. Experimentally the Mueller matrix is measured by Generalized Ellipsometry (GE), which allows for adjustment of the Angle and the Plane of Incidence a beam of electromagnetic radiation makes with respect to a sample surface, as well as rotation of a sample about a perpendicular to said sample surface. Further, during a (GE) measurement different polarization states of the incident light are prepared and their change upon reflection from or transmission through a sample is determined.
Optical Hall Effect (OHE) instruments conduct GE measurements on samples in high quasi-static magnetic fields, and detect the magnetic field induced changes of the Mueller matrix. Though several instruments with partial (OHE) capability are described in the literature, most thereof do not fulfill all desirable criteria for a true (OHE) instrument. For instance, in 1985 Nederpel and Martens, published an article, (see Review of Scientific Instruments, 56,687 (1985)), reported development of a single wavelength (444 nm) magneto-optical ellipsometer for use in the visible spectral range, but the instrument provided only low magnetic fields, (ie. B less than 50 mT). An instrument providing higher magnetic fields with spectroscopic generalized ellipsometry capabilities in the visible spectral range and a vector magnet, (ie. B in the range of 0.4 T) was presented in 2003 by Cerne et al., (see Review of Scientific Instruments, 74, 4755 (2003)). This article presented a magneto-polarimetry instrument which provided a higher magnetic field strength, (ie. B up to 8 T), for use in the mid-infrared spectral range, (ie. spectral lines of a CO2 laser), and in 2004 Padilla et al. developed a terahertz-visible, (ie. 6 to 20000 cm−1 wavelength), magneto-reflectance and transmittance instrument, (ie. a B less than or equal to 9 T), (see Review of Scientific Instruments, 74, 4710, (2004)). While both instruments provide high magnetic fields, and contain polarizers and photo-elastic-modulators, these instruments were not designed to record Mueller matrix data (GE).
A THz time-domain spectroscopy based instrument capable of recording the complex reflection coefficients at magnetic B fields of about 0.5 T was described in 2004 by Ino et al., (see Phys. Rev. B 70, 155101, (2004)). A full 4×4 Mueller matrix in the terahertz-mid-infrared spectral range (20 to 4000 cm−1) can be measured by an instrument described in 2013 by Stanislavchuk et al., (see Review of Scientific Instruments, 84, 023901, (2013)), but there the instrument was not designed for experiments with the sample exposed to external magnetic fields.
The first full (OHE) instrument was developed and demonstrated in 2006 by Inventor herein, Hofmann, (see Review of Scientific Instruments, 77, 63902, (2006)) for the far-infrared (FIR) spectral range (30 to 650 cm−1), which provided magnetic fields up to 6 T and allowed sample temperatures between 4.2 K and room temperature. This first full capability (OHE) instrument has since been successfully used to determine free charge carrier properties including effective mass parameters for a variety of material systems. Later, (OHE) experiments were conducted in the terahertz (THz) spectral range, but were limited to room temperature and low magnetic fields (ie. B fields less than or equal to 1.8 T), and are subject of the invention disclosed herein.
Since the magnitude of the (OHE) depends on the magnetic field strength, higher magnetic fields facilitate the detection of the OHE. Furthermore, the sensitivity to the (OHE) is greatly enhanced by phonon mode coupling, surface guided waves and Fabry-Perot interferences. Since these effects appear from the THz to the mid-infrared (MIR) spectral range, depending on the structure and material of the sample, it can become necessary to extend the spectral range covered by (OHE) instrumentation. An (OHE) instrument for the MIR, for example, can detect the magneto-optic response of free charge carriers enhanced by phonon modes present in the spectral range above 600 cm−1, which applies to many substrate materials, SiC, Al2O3 or GaN, as well as to many materials used for thin films, III-V nitride semiconductors Al1-xGaxN In1-xN Al1-xInxN or In1-xGaxN. In addition, inter-Landau-level transitions can be studied in the MIR spectral range with a MIR (OHE) instrument. The extension to the THz spectral range enables the detection of the (OHE) in samples with low carrier concentrations. Furthermore, the strongest magneto-optic response can be observed at the cyclotron resonance frequency, which typically lies in the microwave/THz spectral range for moderate magnetic fields, (eg. a few Tesla), and effective mass values comparable to the free electron mass.
With the foregoing insight it is noted that the present invention presents an (OHE) instrument that covers an ultra-wide spectral range from 3/cm to 7000/cm, (ie. 0.1-210 THz or 0.4-870 meV), which combines MIR, FIR and THz magneto-optic generalized ellipsometry in a single instrument. This integrated MIR, FIR and THz (OHE) instrument can incorporate a commercially available, closed cycle refrigerated, superconducting 8 Tesla magneto-cryostat sub-system, with four optical ports, providing sample temperatures between T=1.4 K and room temperature. However, the preferred embodiment applies at least one permanent magnet with strength in the range of 0.6 to 1.8 T. The ellipsometer sub-systems used to actually achieve results reported herein, were built in-house and operate in the rotating-analyzer configuration, (a non-limiting election), and are capable of determining the normalized upper 3×3 block of the sample Mueller matrix. Said (OHE) provides insight into free charge carrier properties such as effective mass (m), mobility (u), and carrier concentration (N) of complex and even layered samples. It is noted that the optical Hall Effect (OHE) reveals fundamental symmetry properties of the magneto-optic dielectric tensor.
For insight it is noted that operation of the integrated MIR, FIR and THz (OHE) instrument described was demonstrated by three sample systems. Combined experimental data from the MIR, FIR and THz spectral range of a single epitaxial graphene sample, grown on a 6H-SiC substrate by thermal decomposition were achieved. The MIR (OHE) data of the same epitaxial graphene sample was investigated to demonstrate the operation of the MIR ellipsometer sub-system of the integrated MIR, FIR and THz (OHE) instrument, over the full available magnetic field range of the instrument. The magneto-optic response of free charge carriers and quantum mechanical inter-Landau-level transitions were observed, and their polarization selection rules obtained therefrom noted. A Te-doped, n-type GaAs substrate served as a model system for the FIR spectral range of the FIR/THz ellipsometer sub-system. The (OHE) signal originating from valence band electrons in a bulk material were noted, and the concentration, mobility, and effective mass parameters of the valence band electrons determined. Finally, (OHE) data from an AlGaN/GaN high electron mobility transistor structure (HEMT) from the THz spectral range of the FIR/THz ellipsometer sub-system were achieved and analyzed. The data was recorded at different temperature between T=1.5 K and room temperature, representing the full sample temperature range of the instrument. The results achieved at room temperature were especially important as regards the present invention.
In this Background Section, in what directly follows, dielectric and magneto-optic dielectric tensors are described, a brief theoretical overview on Mueller matrices and GE data-acquisition is given, and general GE data analysis procedures are introduced. In the Detailed Description and Drawing Sections of this Application a description of a relevant, but not-necessarily limiting experimental setup is described, along with data acquisition and data analysis procedures for (OHE) data, and examples of experimental results demonstrating the operation of the integrated MIR, FIR and THz (OHE) instrument are presented and discussed.
Continuing, the evaluation of physically relevant parameters from the (OHE) requires the experimental observation and quantification of the OHE, and a physical model to analyze (OHE) data. Experimentally, the (OHE) is quantified in terms of the Mueller matrix MOHE by employing Generalized Ellipsometry (GE). The physical model which is used to analyze the observed transverse and longitudinal magneto-optic birefringence of the (OHE) is based on the magneto-optic dielectric tensor €oeh(B), which is a function of the slowly varying external magnetic field B. If, among other parameters, the magneto-optic dielectric tensor of a sample is known, experimental Mueller matrices MOHE can be modeled from €oeh(B) using the relationship:MOHE(€oeh(B))Although this equation is in general not invertible analytically, it can be used to determine the magneto-optic dielectric tensor from experimental Mueller matrix data through non-linear model mathematical regression analysis. Dielectric tensors, Mueller matrix calculus, generalized ellipsometry including data acquisition, as well as data analysis are further addressed in this section.Magneto-Optical Dielectric Tensors
The optical response of a sample is here described by the dielectric tensor €. If the dielectric tensor of the sample without a magnetic field is given by €B=0 and the change of the dielectric tensor induced by a magnetic field B is given by €B, the dielectric tensor describing the (OHE), can be expressed as:€(OHE)=€B=0+€B.The magneto-optic permittivity of a material within a given sample described by €B may originate from the response of bound and unbound charge carriers subjected to the magnetic field and the action of the Lorentz force. The magneto-optic response of a sample subjected to the integrated MIR, FIR and THZz (OHE) instrument, and which is addressed herein is represented by a generally anisotropic and nonreciprocal tensor. Thus, the corresponding magneto-optic contributions X+ and X− to the permittivity tensor X=ε−I, (where I is the 3×3 identity matrix), originate from the interaction of right- and left-handed circularly polarized light with the sample, respectively. Without loss of generality, if the magnetic field B is pointing in vector direction indicated by P=ε0χE, it can be described by arranging the electric fields in their circularly polarized eigensystem Ee=(Ex+iEy, Ex−iEy, Ez)=(E+,E−,Ez) by Pe=ε0χeEe=ε0(χ+E+,(χ−E−,0), where i=sqrt{−1} is the imaginary unit. Transforming Pe back into the laboratory system the change of the dielectric tensor induced by the magnetic field takes the form:
                              ɛ          n                =                              1            2                    ⁢                      (                                                                                (                                                                                            x                          ++                                                ⁢                        x                                            -                                        )                                                                                        i                    ⁡                                          (                                              x                        +                                                  -                          x                                                -                                            )                                                                                        0                                                                                                  -                                          i                      (                                                                                                    x                            ++                                                    ⁢                          x                                                -                                                                                                                                                                                x                        ++                                            ⁢                      x                                        -                                                                    0                                                                              0                                                  0                                                  0                                                      )                                              (        3        )            Note, under field inversion B is reversed into −B, the polarizabilities for left- and right-handed circularly polarized light interchange. €B is only diagonal if X+=X−, and otherwise non-diagonal with anti-symmetric off diagonal elements.Classic Dielectric Tensors (Lorentz-Drude Model)
Charges carriers, subject to a slowly varying magnetic field obey the classical Newtonian equation of motion (Lorentz-Drude model):m{umlaut over (x)}+mγ{dot over (x)}+mω02x=qE+q({dot over (x)}×B)  (4)where m, q, μ=qm−1−1, x and ω represent the effective mass tensor, the electric charge, the mobility tensor, the spatial where m, q, μ=qm−1−1, x and ω represent the effective mass tensor, the electric charge, the mobility tensor, the spatial coordinate of the charge carrier and the Eigen-frequency of the un-damped system without external excitation and magnetic field, respectively. For a time harmonic electromagnetic plane wave with an electric field E→E exp(−iωt) with angular frequency ω, the time derivative of the spatial displacement of the charge carrier is {dot over (x)}=v exp(iωt), where v is the velocity of the charge carrier. With j=nqv Eq. 4 reads:
                    E        =                  1          /                                    nq              ⁡                              [                                                      l                    ⁢                                                                                  ⁢                                          m                      /                      q                                        ⁢                                                                                  ⁢                                          ω                      ⁡                                              (                                                                                                            ω                              ⁡                                                              (                                                                                                                                            2                                                                                                                                                                                  0                                                                                                                                      )                                                                                      ⁢                            l                                                    -                                                                                    ω                              2                                                        ⁢                            l                                                    -                                                      i                            ⁢                                                                                                                  ⁢                            ω                            ⁢                                                                                                                  ⁢                            γ                                                                          )                                                              ⁢                    j                                    +                                      (                                          Bx                      ⁢                                                                                          ⁢                      j                                        )                                                  ]                                      .                                              (        5        )            where n is the charge carrier density. With the Levi-Cevita-Symbol €(ijk), (note, in the following equation the Einstein notation is used, and the covariance tensor and contravariance is ignored since all coordinate systems are Cartesian, and the summation is only executed over pairs of lower indices), the conductivity tensor 0−, the dielectric constant €0, and using E=0−1j and the dielectric tensor for charge carriers subject to the external magnetic field B can be expressed as:
                    ɛικ        =                                            nq              2                        /            ɛ                    ⁢                                          ⁢                                                    n                ⁡                                  [                                                            m                      ⁢                                                                                          ⁢                      i                      ⁢                                                                                          ⁢                                              κ                        ⁡                                                  (                                                                                    w                              ⁡                                                              (                                                                                                                                            2                                                                                                                                                                                  0                                                                                                                                      )                                                                                      -                                                          w                              2                                                        -                                                          i                              ⁢                                                                                                                          ⁢                              ω                              ⁢                                                                                                                          ⁢                              γ                              ⁢                                                                                                                          ⁢                              i                              ⁢                                                                                                                          ⁢                              κ                                                                                )                                                                                      -                                          i                      ⁢                                                                                          ⁢                      ωɛ                      ⁢                                                                                          ⁢                      i                      ⁢                                                                                          ⁢                      j                      ⁢                                                                                          ⁢                      κ                      ⁢                                                                                          ⁢                      q                      ⁢                                                                                          ⁢                      B                      ⁢                                                                                          ⁢                      j                                                        ]                                                            -                1                                      .                                              (        6        )            Polar Lattice Vibrations, (Lorentz Oscillator)
For isotropic effective mass tensors the cyclotron frequency can be defined, and for the mass of the vibrating atoms of polar lattice vibrations, the cyclotron frequency is several orders of magnitude smaller than for effective electron masses, and can be neglected for the magnetic fields and spectral ranges discussed in this Specification. Therefore, the dielectric tensor of polar lattice vibrations €L can be approximated using Eq. 6 with B=0. When assuming isotropic effective mass and mobility tensors, the result is a simple harmonic oscillator function with Lorentzian-type broadening. For materials with orthorhombic symmetry and multiple optical excitable lattice vibrations, the dielectric tensor can be diagonalized to:
                              ɛ          L                =                  (                                                                      (                                      ɛ                    ⁢                                          L                      X                                                        )                                                            0                                            0                                                                    0                                                              ɛ                  ⁢                                                                          ⁢                                      L                    Z                                                                              0                                                                    0                                            0                                                              ɛ                  ⁢                                      L                    Z                                                                                )                                    (        7        )            
Where
  Where  ⁢          ⁢      ɛ    ⁡          (                                    L                                                K                              )        ⁢          ⁢  for  ⁢          ⁢      (          k      -              (                              x            1                    ⁢                      y            1                    ⁢                      z            1                          )              )  for (k−(x1y1z1)) is given by:
                                          ɛ            ⁢                                                  ⁢                          L              /              K                                =                      ɛ            ⁢                                                  ⁢            ∞                          ,                              k            ⁢                                                  ⁢                                          π                ⁡                                  (                                                                                    l                                                                                                                                      j                          =                          1                                                                                                      )                                            j                                =                                    1              ⁢                                                          ⁢              ω              ⁢                                                          ⁢              2                        +                          i              ⁢                                                          ⁢              ω              ⁢                                                          ⁢              γ              ⁢                                                          ⁢              L              ⁢                                                          ⁢              0                                      ,                              k            ⁢                                                  ⁢            j                    -                      ω            ⁢                                                  ⁢            2            ⁢                                                  ⁢            L            ⁢                                                  ⁢            0                          ,                              k            ⁢                                                  ⁢                          j              /              ω                        ⁢                                                  ⁢            2                    +                      i            ⁢                                                  ⁢            ω            ⁢                                                  ⁢            γ            ⁢                                                  ⁢            T            ⁢                                                  ⁢            0                          ,                              k            ⁢                                                  ⁢            j                    -                      ω            ⁢                                                  ⁢            2            ⁢                                                  ⁢            T            ⁢                                                  ⁢            0                          ,        kj                            (        8        )            Where ωL0,j,k, γL0,j,k, ωT0,j,k and γT0,j,k denote the k(x,y,z) component of the frequency and the broadening values of the jth longitudinal optical (LO) and transverse optical (TO) phonon modes, respectively, while the index j runs over 1 modes. Further details can be found in Hofmann et al., Applied Physical Letters 88, 042105 (2006); Barker, Phys. Rev., 136, A1290 (1964); Berryman et al. Phys. Rev. 174, 791, (1968); Gervais et al., J. Phys. C 7, 2374, (1974); Hofmann et al., Phys. Rev. B, 66, 19504 1 (2002) 1 and a discussion of the requirements to broadening parameters, such as Im
  (      ɛ    ⁢          L      K        )greater than or equal to 0.0 are found in Kasic et al., Phys. Rev. B 61,7365, (2000).Free Charges Carriers (Extended Drude Model)
For free charge carriers no restoring force is present and the Eigen-frequency of the system is ω0=0. For isotropic effective mass and conductivity tensors, and magnetic fields aligned along the z-axis Eq. 6 can be written in the form for B=0.
                                                        ω              ⁡                              (                                                                            2                                                                                                  p                                                                      )                                      /                          ω              ⁡                              (                                  ω                  =                                      i                    ⁢                                                                                  ⁢                    γ                                                  )                                              ⁢          l                =                            ⁢          l          ⁢                      :                                              (        9        )            where ωP is the plasma frequency, and €D is permittivity function of the isotropic Drude dielectric function. The magneto-optic contribution to the dielectric tensor €D for isotropic effective masses and conductivities can be expressed, using Eq. 3, through polarizability functions for right- and left-handed circularly polarized light:x±=−(€D)/(1±(ω+jγ/ωc)),  (10)where ωc=q B/m is the isotropic cyclotron frequency.Non-Classic Dielectric Tensors (Inter-Landau-Level Transitions)
The permittivity tensor describing the contribution of a series of inter-Landau-level transitions to the dielectric tensor can be approximated by a sum of Lorentz oscillators. The quantities in Eq. (3) are then expressed by:
                                          x            ±                    =                                                    e                                                      ±                                                                                  ⁢                    i                                    ⁢                                                                          ⁢                  ϕ                                            ⁡                              (                                  (                                                                                    ∑                                                                                                            k                                                                              )                                )                                      ⁢                                          (                                  A                  ⁢                                                                          ⁢                  κ                                )                            /                              (                                                      ω                    2                                    -                                      ω                                          (                                                                        2                          /                          0                                                ·                        κ                                            )                                                        -                                      i                    ⁢                                                                                  ⁢                    γ                    ⁢                                                                                  ⁢                    κ                    ⁢                                                                                  ⁢                    ω                                                  )                                                    ,                            (        11        )            where AK, ω0, an K are amplitude, transition energy, and broadening parameter of the Kth inter Landau-level transition, respectively, which in general depend on the magnetic field. The phase factor was introduced here to describe transition, respectively, which in general depend on the magnetic field. The phase factor was introduced here to describe the experimentally observed line shapes of all Mueller matrix elements.
For inter-Landau-level transitions in graphite or bi-layer graphene we find φ=π/4, and for inter-Landau-level transitions in single layer graphene φ=0.
Note that for φ=0, the polarizabilities for left and right handed circularly polarized light are equal, (χ+=χ−), and
  ɛ  ⁢      LL    B  is diagonal.Mueller Matrix Calculus, GE and Data Acquisition
Generalized ellipsometry (GE) extends standard, isotropic ellipsometry (SE) to arbitrary anisotropic and depolarizing samples by including rotation about a perpendicular to a sample surface, and can reveal the complex 3×3 dielectric tensor of the material investigated. This section describes the Jones vector/Mueller matrix formalism used in GE, aspects of Mueller matrix and (OHE) data, and the acquisition of Mueller matrix data.
Stokes Vector/Mueller Matrix Calculus
The real-valued Stokes vector S has four components, carries the dimensions of intensity, and can quantify any polarization state of plane electromagnetic waves. If expressed in terms of the p- and s-coordinate system Stokes vector S has four components, carries the units of intensity, and can quantify any polarization state of plane electromagnetic waves. I−45 and S4=Iσ+−Iσ−, with Ip, Is, I45 I−45, Iσ+ and Iσ− being intensities for the p−. s− +45°, −45°, right and left handed circularly polarized light components, respectively. (See R. M. Azzam and N. M. Bashara, “Ellipsometry and Polarized Light”, North-Holland Publ. Co., Amsterdam, (1984)).
The real-valued 4×4 Mueller matrix M describes the change of electromagnetic plane wave properties (intensity, polarization state), expressed by a Stokes vector S, upon change of the coordinate system or the interaction with a sample, optical element, or any other matter:si(out)=Σi=13MijSi(in), (=1 . . . 4),  (12)Where S(out) and S(in) denote the Stokes vectors of the electromagnetic plane wave before and after the change of the coordinate system, or an interaction with a sample, respectively. Note that all Mueller matrix elements of the GE data discussed in this paper, are normalized by the element M11, therefore Mij is less than or equal to and M11=1.Mueller Matrix and (OHE) Data
The Mueller matrix can be decomposed in 4 sub-matrices, where the matrix elements of the two off-diagonal-blocks:
      [                                        M            ⁢                                                  ⁢            13                                                M            ⁢                                                  ⁢            14                                                            M            ⁢                                                  ⁢            23                                                M            ⁢                                                  ⁢            24                                ]    =      [                                        M            ⁢                                                  ⁢            21                                                M            ⁢                                                  ⁢            14                                                            M            ⁢                                                  ⁢            41                                                M            ⁢                                                  ⁢            42                                ]  only deviate from zero if p− to s− polarization mode conversion appears, while the matrix elements in the two on-diagonal-blocks:
      [                                        M            ⁢                                                  ⁢            13                                                M            ⁢                                                  ⁢            14                                                            M            ⁢                                                  ⁢            23                                                M            ⁢                                                  ⁢            24                                ]    =      [                                        M            ⁢                                                  ⁢            21                                                M            ⁢                                                  ⁢            14                                                            M            ⁢                                                  ⁢            41                                                M            ⁢                                                  ⁢            42                                ]  mainly contain information about p− to s− polarization mode conversion, while the matrix elements in the two on-diagonal-blocks mainly contain information about p− s− polarization mode conserving processes. It is to be appreciated that p− to p− polarization mode conversion is defined as the transfer of energy from the p− polarized channel of an electromagnetic plane wave to the s− polarized channel, or vice versa. Polarization mode conversion can appear when the p− s− coordinate system is different for Sin and Sout, or, when a sample shows birefringence, for example. In particular, polarization mode conversion appears if the dielectric tensor of a sample possesses non-vanishing off-diagonal elements. Therefore, in Mueller matrix data from optically isotropic samples, ideally all off-diagonal-block elements vanish, while, for example, magneto-optic birefringence can cause non-zero off-diagonal-block elements in the Mueller matrix.
Here, we define (OHE) data as Mueller matrix data from an (OHE) experiment [Eq. 1] with magnetic field +/−B and the corresponding to the zero field dataset:
                              M          OHE          ±                =                  M          ⁡                      (                                          ɛ                                  B                  =                  0                                            +                              ɛ                                  +                                      /                                          -                      B                                                                                            )                                              (        13        )            
                              δ          ⁢                                          ⁢                      M            ±                          =                                            M              OHE              ±                        -                          M              0                                =                      Δ            ⁢                                                  ⁢                          M              ⁡                              (                                                      ɛ                                          B                      =                      0                                                        +                                      ɛ                                          ±                      B                                                                      )                                                                        (        14        )            where M0=M(εB=0) is the Mueller matrix of the zero field experiment, and ΔM(εB=0;εB) is the magnetic field induced change of the Mueller matrix. This form of presentation is in particular advantageous in case the magnetic field causes only small changes in the Mueller matrix, and provides improved sensitivity to magnetic field dependent model parameters during data analysis. Another form of presentation for derived (OHE) data is:
                                                        δ              ⁢                                                          ⁢                              M                +                                      ±                          d              ⁢                                                          ⁢                              M                -                                              =                                    Δ              ⁢                                                          ⁢                              M                ⁡                                  (                                                            ɛ                                              B                        =                        n                                                              ;                                          ɛ                                              +                        B                                                                              )                                                      ±                          Δ              ⁢                                                          ⁢                              M                ⁡                                  (                                                            ɛ                                              B                        =                        n                                                              ;                                          ɛ                                              -                        B                                                                              )                                                                    ,                            (        15        )            that can be used to inspect symmetry properties of magneto-optic Mueller matrix data, and can help to improve the sensitivity to magnetic field dependent model parameters during data analysis.Mueller Matrix Data Acquisition (GE)
Spectroscopic ellipsometers can be categorized according to their polarization optical components and operation principles, where different subsets of Mueller matrix elements may be accessible. For example Spectroscopic ellipsometers can be classified into two categories: (i) polarizer-sample+rotating analyzer ellipsometers (PSAR) or rotating-polarizer+sample-analyzer (PRSA) configurations, capable of measuring the upper left 3×3 block of the Mueller matrix; and (ii) rotating compensator(s) ellipsometers (RCE) in polarizer-sample-rotating-compensator-analyzer (PSCRA) or polarizer-rotating-compensator-sample-analyzer (PCRSA) configuration, capable of measuring the upper left 3×4 or 4×3 block of the Mueller matrix, respectively.
Mathematically all ellipsometers, can be described by the ordered multiplication of Mueller matrices, corresponding to their consecutive optical elements. The Mueller matrices of a polarizer (P), analyzer (A), compensator (δ) with phase shift, coordinate rotation along beam path (Rθ) by an angle θ, and of the sample (M) are given by:
                                                        P              =                              A                =                                                      1                    2                                    ⁡                                      [                                                                                            0                                                                          0                                                                          1                                                                          1                                                                                                                      0                                                                          0                                                                          1                                                                          1                                                                                                                      0                                                                          0                                                                          0                                                                          0                                                                                                                      0                                                                          0                                                                          0                                                                          0                                                                                      ]                                                                                                                          R                ⁡                                  (                  θ                  )                                            =                              [                                                                            1                                                              0                                                              0                                                              0                                                                                                  0                                                                                      cos                        ⁡                                                  (                                                      2                            ⁢                                                                                                                  ⁢                                                          θ                              j                                                                                )                                                                                                                                    sin                        ⁡                                                  (                                                      2                            ⁢                                                                                                                  ⁢                                                          θ                              j                                                                                )                                                                                                            0                                                                                                  0                                                                                      -                                                  sin                          ⁡                                                      (                                                          2                              ⁢                                                                                                                          ⁢                                                              θ                                j                                                                                      )                                                                                                                                                              cos                        ⁡                                                  (                                                      2                            ⁢                                                                                                                  ⁢                                                          θ                              j                                                                                )                                                                                                            0                                                                                                  0                                                              0                                                              0                                                              1                                                                      ]                                                                                                        C                ⁡                                  (                  δ                  )                                            =                              [                                                                            1                                                              0                                                              0                                                              0                                                                                                  0                                                              1                                                              0                                                              0                                                                                                  0                                                              0                                                                                      cos                        ⁡                                                  (                          δ                          )                                                                                                                                    -                                                  sin                          ⁡                                                      (                            δ                            )                                                                                                                                                                          0                                                              0                                                                                      sin                        ⁡                                                  (                          δ                          )                                                                                                                                    cos                        ⁡                                                  (                          δ                          )                                                                                                                    ]                                                                        M              =                              [                                                                                                    M                        ⁢                                                                                                  ⁢                        11                                                                                                            M                        ⁢                                                                                                  ⁢                        12                                                                                                            M                        ⁢                                                                                                  ⁢                        13                                                                                                            M                        ⁢                                                                                                  ⁢                        14                                                                                                                                                M                        ⁢                                                                                                  ⁢                        21                                                                                                            M                        ⁢                                                                                                  ⁢                        22                                                                                                            M                        ⁢                                                                                                  ⁢                        23                                                                                                            M                        ⁢                                                                                                  ⁢                        24                                                                                                                                                M                        ⁢                                                                                                  ⁢                        31                                                                                                            M                        ⁢                                                                                                  ⁢                        32                                                                                                            M                        ⁢                                                                                                  ⁢                        33                                                                                                            M                        ⁢                                                                                                  ⁢                        34                                                                                                                                                M                        ⁢                                                                                                  ⁢                        41                                                                                                            M                        ⁢                                                                                                  ⁢                        42                                                                                                            M                        ⁢                                                                                                  ⁢                        43                                                                                                            M                        ⁢                                                                                                  ⁢                        44                                                                                            ]                                                                        (        16        )            respectively. Execution of the matrix multiplication characteristic for the corresponding ellipsometer type shows that, due to the rotation of optical elements, the measured intensity at the detector is typically sinusoidal. Fourier analysis of the detector signal provides Fourier coefficients, which are used to determine the Mueller matrix of the sample.Data Analysis
Ellipsometry is generally an indirect experimental technique. Therefore, in general, ellipsometric data analysis invokes model calculations to determine physical parameters in dielectric tensors or the thickness of layers, for instance. Sequences of homogeneous layers with smooth and parallel interfaces are assumed in order to calculate the propagation of light through a layered sample, by the 4×4 matrix formalism. To best match the generated data with experimental results, parameters with significance physical model parameter in dielectric tensors, layer thicknesses etc. are varied and Mueller matrix data is calculated for all spectral data points, angles of incidence and magnetic fields. During the mean square error (MSE) regression, the generated Mueller matrix data Mi,j,kG is compared with the experimental Mueller matrix data Mi,j,kB and their match is quantified by the MSE:
  MSE  =                              (                      1                                          9                ⁢                                                                  ⁢                S                            -              k                                )                ⁢                              ∑                          i              =              1                        4                    ⁢                                    ∑                              j                =                1                            4                        ⁢                          ∑                              k                =                1                            5                                            ⁢          (              (                              M            ⁡                          (                              E                                  i                  ,                  j                  ,                  k                                            )                                -                                                    M                ⁡                                  (                                      G                                          i                      ,                      j                      ,                      k                                                        )                                            /              6                        ⁢                                                  ⁢                          M              ⁡                              (                                  E                                      i                    ,                    j                    ,                    k                                                  )                                      ⁢                                                  ⁢            2.                              where S, K and σMi,j,kG denotes the total number of spectral data points, the total number of parameters varied during the non-linear regression process, the number experimentally determined columns and rows of the Mueller matrix and the standard viation of M{I,j,k}, obtained during the experiment, respectively. For fast convergence of the MSE regression, the Levenberg-Marquardt fitting algorithm is used. The MSE regression is interrupted when the decrease in the MSE is smaller than a set threshold and the determined parameters are considered as best model parameters. The sensitivity and possible correlation of the varied parameters is checked and, if necessary, the model is changed and the process is repeated. Eventually values for parameters in the mathematical model of the sample being characterized are arrived at and represent very reliable insight to actual physical values.
It is also mentioned, that is some special cases ellipsometry can provide results that allow direct analytical solutions for such as concentration and mobility of charge carriers based on off diagonal Jones or Mueller Matrix element value slopes, as a function of an applied magnetic field. While the typical approach to determining at least one of the free charge carrier longitudinal and/or transversal effective masses, and/or concentration, and/or mobility and/or type from said anisotropic values for said at least a partial Jones or Mueller Matrix determining involves use of mathematical regression onto a model of the sample and ellipsometer or polarimeter system used to evaluate the Jones or Mueller Matrix elements, it is possible under certain circumstances to arrive at concentration and mobility of carriers by a direct calculation. This is because in the situation wherein the optical path length in a sample, including associated, (eg. Fabry-Perot), cavity forming elements, is a multiple of the wavelength, (ie. Fabry-Perot resonance is achieved, there is a linear relationship between applied magnetic field and slope in Off-diagonal Matrix elements. The slope is different for each of the off-diagonal Matrix elements determined, but in all cases depends only on mobility and concentration of charge carriers. Further, while this determination involves use of slopes in two such off-diagonal Matrix elements as the applied magnetic field is ramped up, it is generally considered that the slope be determined at two such applied magnetic fields. However, as the off-diagonal Matrix elements vanish when the applied magnetic field is zero, a single applied field measurement can sufficient to determine the necessary slopes.
THz Time-Domain Spectroscopy Based Ellipsometry Beside THz frequency-domain spectroscopy based ellipsometry discussed in this Specification, ellipsometry and magneto-optic ellipsometry can be conducted using THz time-domain spectroscopy (THz-TDS), see Sakai, “Terahertz Optoelectronic” Springer-Verlag, (2005}. Typically THz-TDS is based on the Fourier transformation of the time resolved signal of ultra-short (picosecond) laser pulses, revealing the THz spectrum. THz-TDS was developed in the 1980s, and had had its practical breakthrough in the 1990s. THz-TDS has since been used to study the birefringence of a variety of materials in the THz range, as well as the THz magneto-transmittance for a variety of semiconductors. Polarization sensitive THz magneto-transmittance based on THz-TDS were reported in 1997. THz ellipsometry based on THz-TDS was reported by the Hangyo-group in 2001, Nagashima & Hangyo, (see App. Phys. Lett., 79, 3017, (2001), and Matsumoto et al., J. J. APP. Phys. 48, (2009) and Matsumoto et al., Optics Letters, Vol. 36, No. 2, (Jan. 15, 2011), and in 2012 by Neshat Op. Soc., Vol 20, No. 27, (2012)). THz-TDS magneto-ellipsometry measurements were reported by Ino et al., (see Phys. Rev. B, 70, (2004)), but external magnetic fields were limited to B approximately 0.5 T and Mueller matrix capabilities of the instrument were not demonstrated.Search of Patents and Published Applications
A Search of the USPTO Database was conducted for the terms “Optical Hall Effect” (OHE) and independently, (1) Ellipsometer, (2) Polarimeter, (3) Terahertz, (4) THz, in both Issued Patent and Published Application categories. No hits were found. When “Optical Hall Effect” was Searched on its own, without an accompanying additional term, many hits were obtained in both categories. However, said hits seem to be referring to “in the alternative” type systems. That is, an invention can use Hall Effect or Optical sensing etc. to arrive at a desired result. While not particularly relevant to the Invention Claimed herein, a few known Patents that describe Terahertz Ellipsometer Systems or the like are U.S. Pat. No. 8,169,611 8,416,408; 8,488,119; 8,705,032 and 8,736,838 to Herzinger or Herzinger et al., and which are assigned to the J.A. Woollam Co. Inc., some in conjunction with the Board of Regents of the University of Nebraska. It is also noted that while the present invention has generally been practiced by the Inventors herein using a Rotating Analyzer Ellipsometer configuration, any Ellispometer configuration can be applied including Rotating Polarizer or Rotating Compensator, and combinations thereof. As well, Modulation Element Ellipsometers can also be employed and should be considered within the Claims if not otherwise excluded by Claim language. Although not specifically directed to Infrared or Terahertz wavelength ranges, An example of a Patent covering such a Modulation Element configuration is U.S. Pat. No. 5,657,126 to Duchamrme et al.
Finally in this Background Section of the Specification, it is noted that terminology used in the Claims regarding Vacuum Ultraviolet, Mid-Ultraviolet, Near-Ultraviolet, Visible, Near-Infrared, Mid-Infrared, Far-Infrared, (eg. some being examples of Fourier Transform Infrared) and Terahertz wavelength ranges can be roughly defined as:                VUV (eg. 0.01-0.2 um);        MUV (eg. 0.2-0.3 um);        NUV (eg. 0.3-0.4 um);        Visible (eg. 0.4-0.75 um);        NIR; (eg. 0.75-3 um);        MIR; (eg. 3-30 um);        FIR; (eg. 30-350 um); and        THz (eg. 350→1000 um).        
The relevant literature is not absolutely consistent in said definitions however and differences in how various publications define said ranges should not be interpreted to be significant in disclosure of the present invention.
For insight, assuming a classical Newtonian equation of motion with Lorentz force contribution due to the presence of an external static magnetic field (i.e., no quantum phenomena), the dielectric function tensor can be expressed as in equation 6 herein, (eg. the Lorentz-Drude model). For the common case of free-charge carriers, we assume no restoring force, thus w0=0 (Drude Model). The off-diagonal dielectric function tensor elements which constitute the optical Hall effect (birefringence in the presence of magnetic fields, i.e., non-zero off-diagonal MM elements), show significant contributions only in the spectral ranges where we see free-charge carrier contribution also without magnetic field. Even for highly doped semiconductors those contributions are limited to the long-wavelength range or more precisely to wavelength longer than the visible spectral range. For 2D materials with quantum mechanical effects, we observe resonances in the MM data related to discrete transitions between confined electronic states (so-called Landau level transitions). Even in materials with extremely high mobility such as graphene, at ultra-low temperature of 1.5K, and with strong magnetic field of 8T, observable transitions in the MM data are limited to wavelengths longer than 3 um.
It is also pointed out that even though possible effects might not be governed by the exact same equations as in the IR spectral range, there might still be magnetic field induced effects in the UV and VUV spectral range described by a different dielectric function tensor.
A recent article by Knight et al. titled “In Situ Terahertz optical Hall Effect Measurement of Ambient Effects on Free Charge Carrier Properties of Epitaxial Graphene”, Scientific Reports 7:5151, Jul. 11, 2017 is identified. This article demonstrates how the atmospheric content at the surface of epitaxial graphene can have effects, observable in Mueller Matrix Elements related to free charge carrier properties.
Even in view of the prior art, need remains for improved systems and methods of their use that allow the Optical Hall Effect (OHE) to be monitored at room temperature and relatively low Tesla field strengths provided by small permanent magnets.